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The class of Polish spaces has several universality properties, which show that there is no loss of generality in considering Polish spaces of certain restricted forms.
* Every Polish space is [[homeomorphic]] to a [[Gδ space|''G''<sub>δ</sub>
* Every Polish space is obtained as a continuous image of Baire space; in fact every Polish space is the image of a continuous bijection defined on a closed subset of Baire space. Similarly, every compact Polish space is a continuous image of Cantor space.
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=== Borel hierarchy ===
Each Borel set of a Polish space is classified in the '''[[Borel hierarchy]]''' based on how many times the operations of countable union and complementation must be used to obtain the set, beginning from open sets. The classification is in terms of [[countable set|countable]] [[ordinal number]]s. For each nonzero countable ordinal ''α'' there are classes <math>\mathbf{\Sigma}^0_\alpha</math>, <math>\mathbf{\Pi}^0_\alpha</math>, and <math>\mathbf{\Delta}^0_\alpha</math>.
* Every open set is declared to be <math>\mathbf{\Sigma}^0_1</math>.
* A set is declared to be <math>\mathbf{\Pi}^0_\alpha</math> if and only if its complement is <math>\mathbf{\Sigma}^0_\alpha</math>.
* A set ''A'' is declared to be <math>\mathbf{\Sigma}^0_\delta</math>, ''δ'' > 1, if there is a sequence ⟨ ''A''<sub>''i''</sub> ⟩ of sets, each of which is <math>\mathbf{\Pi}^0_{\lambda(i)}</math> for some ''λ''(''i'') < ''δ'', such that <math>A = \bigcup A_i</math>.
* A set is <math>\mathbf{\Delta}^0_\alpha</math> if and only if it is both <math>\mathbf{\Sigma}^0_\alpha</math> and <math>\mathbf{\Pi}^0_\alpha</math>.
A theorem shows that any set that is <math>\mathbf{\Sigma}^0_\alpha</math> or <math>\mathbf{\Pi}^0_\alpha</math> is <math>\mathbf{\Delta}^0_{\alpha + 1}</math>, and any <math>\mathbf{\Delta}^0_\beta</math> set is both <math>\mathbf{\Sigma}^0_\alpha</math> and <math>\mathbf{\Pi}^0_\alpha</math> for all ''α'' > ''β''. Thus the hierarchy has the following structure, where arrows indicate inclusion.
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<math>
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=== Regularity properties of Borel sets ===
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* A set is <math>\mathbf{\Delta}^1_{n}</math> if it is both <math>\mathbf{\Pi}^1_n</math> and <math>\mathbf{\Sigma}^1_n</math> .
As with the Borel hierarchy, for each ''n'', any <math>\mathbf{\Delta}^1_n</math> set is both <math>\mathbf{\Sigma}^1_{n+1}</math> and <math>\mathbf{\Pi}^1_{n+1}
The properties of the projective sets are not completely determined by ZFC. Under the assumption [[axiom of constructibility|''V = L'']], not all projective sets have the perfect set property or the property of Baire. However, under the assumption of [[projective determinacy]], all projective sets have both the perfect set property and the property of Baire. This is related to the fact that ZFC proves [[Borel determinacy]], but not projective determinacy.
There are also generic extensions of <math>L</math> for any natural number <math>n>2</math> in which <math>\mathcal P(\omega)\cap L</math> consists of all the lightface <math>\Delta^1_n</math> subsets of <math>\omega</math>.<ref>V. Kanovei, V. Lyubetsky, "[https://www.mdpi.com/2227-7390/8/9/1477 On the <math>\Delta^1_n</math> problem of Harvey Friedman]. In ''Mathematical Logic and its Applications'' (2020), DOI [https://doi.org/10.3390/math8091477 10.3380/math8091477].</ref>
More generally, the entire collection of sets of elements of a Polish space ''X'' can be grouped into equivalence classes, known as [[Wadge degree]]s, that generalize the projective hierarchy. These degrees are ordered in the [[Wadge hierarchy]]. The [[axiom of determinacy]] implies that the Wadge hierarchy on any Polish space is well-founded and of length [[Θ (set theory)|Θ]], with structure extending the projective hierarchy.
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==References==
* {{cite book | authorlink=Alexander Kechris | author=Kechris, Alexander S. | title=Classical Descriptive Set Theory | url=https://archive.org/details/classicaldescrip0000kech | url-access=registration | publisher=Springer-Verlag | year=1994 | isbn=0-387-94374-9}}
* {{cite book | authorlink=Yiannis N. Moschovakis | author=Moschovakis, Yiannis N. | title=Descriptive Set Theory|url=
===Citations===
{{Reflist}}
== External links ==
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